Qualitative Analysis of Autonomous ODE
Consider the autonomous ode:
The differential equation gives a formula for the slope. In this example, the slope just depends on the independent variable. If the function passes through the point (1,2), the slope is -8. The slope is independent of t, so the slope is constant if y is constant. The following table lists the slope at several y points.
The direction field consists of small lines which characterize the slope of y(t) at each point in the t-y plane. Here is the direction field for our example.
For example, if y(t) passes through the point (1,2) it should have a slope of -8. This is shown by the small line with large negative slope. If y(t) passes through (1,0.5), the slope is 1, and this shown by the line with moderate positive slope.
The direction field gives us a rough idea about solutions to the differential equation, since solutions to the differential equation are tangent to the small slope lines.